Guillaume Bolduc · Preprint · 2026-10-07 · v0.3
Abstract
We study a nominal admission controller with a nonempty protected cohort and finitely many shielded newcomers. Complete coherent residual tables centered on the protected cohort cancel all shielded physical residuals from its mean acceleration. Protected departures and gated promotions have explicit position and velocity coordinate maps. Each population or role event invalidates the preceding residual epoch and uses a bounded zero-correction waiting law. With finitely many born incarnations, at most one promotion per incarnation, the stated delayed-priority contracts, and the parent factored-gain small-age condition, a final continued shield phase captures every remaining newcomer. A finite group release is then admissible, and final relative geometry recovers with position and velocity retained at the switch. Uniform protected-relative and event-seam bounds require an explicit family of uniformly bounded histories, service, forcing, event counts and stability margins; capture time and newcomer effort may grow with initial distance. We derive the final group-release mean response and distinguish summable limiting shifts from summable whole-response envelopes. Proofs concern exact nominal dynamics. New finite rational checks corroborate selected algebra only; no online release detector, collision, actuator, radio, rigid-body or perpetual-churn result is established.
Keywords: formation control · delayed feedback · population changes · staged admission
Guillaume Bolduc · Preprint · 2026-10-06 · v01
Abstract
We study a sampled common-translation latch coupled to equal-gain nominal double integrators. The sampled branches follow the recovery callbacks in the Flying Cats version cited by the parent article: settled bodies are skipped, recomputation precedes a possible priority update, and the returning branch holds the translation while its timer advances. For a fixed finite cohort, convergent goals and vanishing disturbance integrals on fixed sampling cells, exact nominal flow and every sequence of holds or recomputations produce vanishing tracking error and velocity. The source-bound guard consequently stops recomputing after finitely many samples. The endpoint contains the actual frozen translation. We prove augmented translation equivariance, well-posedness, a last-recomputation error formula, and separate timer, impulse and bounded-residual statements. Complete-cohort information and a common clock are explicit resources. The result does not establish a rigid-body, airflow or floating-point realization.
Keywords: formation control · sampled feedback · common translation · hybrid systems · disturbance recovery
Guillaume Bolduc · Preprint · 2026-10-06 · v05
Abstract
We study recovery of relative three-dimensional geometry without an absolute restoring force on the formation translation. A least-squares translation removes the mean target error. For equal-gain double integrators, the resulting common mode is a damped integrator and the relative modes are stable second-order systems. A causal realization uses complete same-time origin snapshots, a shared commit clock and a current local residual innovation. Its estimation errors sum to zero exactly, although individual estimates are stale. A sufficient small-age condition proves boundedness and recovery for the coupled sampled feedback under actual communication contracts, including directional and state-dependent delays. An actual delayed priority process supplies convergent inputs under bounded use ages and joint connectivity of consumed influence. Finite cohort changes retain surviving physical and estimator states and preserve departed bodies under a separate law. Shared anchor tables cancel from terminal relative shape; fixed local profile tables yield an explicit priority-dependent shape and a nonconstancy criterion. Finite-variation velocity impulses, conditional active-body tubes and an exact nominal event map are treated separately. Finite checks illustrate the formulas. No rigid-body implementation, continuous trajectory safety or novelty claim is inferred from them.
Keywords: formation control · communication delay · consensus · disturbance recovery · open multi-agent systems
Guillaume Bolduc · Preprint · 2026-10-04 · v09 correction 01
Abstract
We study a deterministic conditional-query problem in which a sampled index is encoded before a later query, the current index may change, and every surviving input-dependent distinction belongs to a finite historical alphabet. On a logarithmic probability grid with \(P\) points, displacement bound \(A\), spacing \(h\), and at most \(K\) historical states, we obtain the exact worst Euclidean query error \(\tanh(hD_*/4)/\sqrt2\). The optimal expanded-fiber diameter is \(D_*=P-1\) when \(2A\ge P-1\), and \(D_*=2A+\lceil(P-2A)/K\rceil-1\) otherwise. Consecutive encoder blocks and query-dependent midpoint responses attain the bound. Rational geometric parameters give a bit-machine implementation with separately counted workspace and output-precision error. A two-slot periodic construction retains bounded-age codes under correct fresh acquisition, bounded acknowledged service and initialized-only scoring. After a finite attack on stored symbols or forward payload fields, clean overwrite restores current-task accuracy while acquisition, executable encoder/decoder and control remain correct. A separate one-hop replicated crash-erasure model admits an exact period-endpoint law under all-copy lifetime, full-drain and no-side-channel contracts. Quadratic objectives transfer the query law to optimizer-estimation error. A normalized family expresses the conditional error in terms of retained mass. The results have distinct information, timing and fault assumptions; they do not establish a general network resource optimum or Byzantine source/control recovery.
Keywords: information constraints · membership changes · communication delay · deterministic minimax estimation
Guillaume Bolduc · Preprint · 2026-10-03 · v0.6
Abstract
We study a sampled reference controller on a finite set of nominal double integrators with changing logical participation. A correct reference service, coherent delayed physical tables, a common clock, and a constrained translation latch are specified as resources. An error estimate accounts for disturbance integrals, goal changes, admission state, snapshot age, and projection error. A weighted event budget gives a tracking bound without requiring participation eventually to become fixed. Separate convex-cell assumptions give continuous separation, speed, and acceleration bounds for all bodies, including inactive bodies. Under a supplied finite service discipline, a delayed-observation gate permits repeated logical reinitialization and readmission of the same bodies without accumulating bodies or workspace. Uniform operational storage additionally requires bounded message production, frame sizes, lifetime, and serviced queues. A convergent quiet tail gives eventual latch freeze under the stated contraction condition. These are conditional results for fixed task coordinates, reserved operating cells, exact local feedback, and correct services; numerical and physical realization remain separate questions.
Keywords: sampled tracking · changing membership · double integrators · event budgets · translation latch · bounded storage
Guillaume Bolduc · Preprint · 2026-10-03 · v02
Abstract
We study a scalar two-agent asynchronous averaging process with bounded discrete communication age. At each primitive action, one agent replaces its current value by a fixed convex combination of its own current value and a delayed value of the other agent, while the other agent holds. A maximum age \(D\) gives an augmented history of dimension \(M=2(D+1)\). Every admissible infinite future that reaches consensus induces a linear functional on that history. Finite-word observability and minimal realization are established switched-system concepts. Here the output family consists of asymptotic endpoints; its static row-space interpretation is elementary and is not claimed as new. The contribution is a protocol-specific rank theorem. For every \(D\ge 0\) and every self-weight \(\theta\in(0,1)\), we construct \(2(D+1)\) admissible period-two futures whose consensus endpoint functionals are linearly independent. The future-functional span is therefore the complete dual of the augmented history space. Consequently, no nontrivial linear historical compression can reproduce the consensus endpoint for every admissible future. The same scalar lower bound holds for continuously differentiable encoders and decoders on a nonempty open set of independently initialized histories. For complete probability rows with \(m\ge2\) objective coordinates, the corresponding intrinsic initialized-history dimension is \(M(m-1)\). These counts do not automatically apply to histories constrained by a preceding lawful execution. Finally, equality of consensus endpoints under every admissible future is equivalent to equality of the complete augmented histories; an affine intervention can therefore achieve schedule-robust restoration only by exactly repairing the full history. Exact rational computations corroborate the witness construction for delays through 12 and six rational self-weights. The result concerns the declared two-agent recurrence. It does not characterize larger graphs, packet availability, nonlinear discontinuous encodings, hardware, or publication priority.
Keywords: asynchronous consensus · communication delay · historical state · observability · restoration
Guillaume Bolduc · Preprint · 2026-10-03 · v0.6
Abstract
We study restoration of probability-valued priorities after finite changes in network membership. For fixed uniform or unequal directional communication delays, the endpoint depends on the complete initialized history. We derive the exact returned endpoint after one admission, finite mixing and proportional departure. Priority restoration is an affine equation in the admitted conditional preference; restoration of a three-dimensional objective centre is its projection through the anchor matrix. A causal packet protocol preserves origin-dependent historical transport, uses reception ordinals to order simultaneous readiness, and yields bounded-age stochastic dynamics after finite membership changes. Under bounded joint connectivity and positive current self-weights, those dynamics converge with an explicit sufficient contraction bound. Restoration over unknown admissible futures is characterized by finitely many endpoint functionals; a finite polynomial characterization additionally requires positive-polynomial initial and bootstrap data. We prove historical survivor-query memory bounds in distinct smooth-state and deterministic finite-bit models. A reference service retaining unmixed rows provides eventual restoration under a modified update law. Finally, a double-integrator controller converts priority convergence into point rendezvous or separated formation convergence. A staged finite-pool construction parks retired bodies in reserved finite-depth berths, while preserving all-body separation and speed and acceleration bounds. All conclusions retain their stated information, timing, storage and geometric assumptions.
Keywords: priority consensus · communication delays · historical restoration · dynamic membership · rendezvous · formation control
Guillaume Bolduc · Preprint · 2026-10-03 · v01
Abstract
A standard live admission can pull every incumbent by a distance proportional to a newcomer's initial separation, despite physical continuity at the event. We give a separate admission law that projects the complete residual table onto incumbent-centered and newcomer-tracking components. The newcomer's physical residual cancels exactly from incumbent feedback while its priority row participates in the actual delayed network. Under the parent paper's coherent-history and small-age assumptions, incumbent common translation obeys its pre-admission free-mode equation, the newcomer approaches the prescribed offset geometry, and a later history-gated switch recovers the ordinary full-cohort dynamics. The incumbent displacement and final common-mode seam have bounds independent of the newcomer's initial absolute distance, although capture time and newcomer actuation do not. We retain the parent disturbance-recovery theorem and the existing membership and delayed-priority results as dependencies, rather than reproving them.
Keywords: live admission · common-mode shielding · delayed priority networks · coherent history · formation control
Guillaume Bolduc · Preprint · 2026-09-11 · v01
Abstract
This note isolates the fixed-epoch averaging result used as Lemma~2.1 of Priority-State Dynamics in Open Multi-Agent Systems: Dynamic Membership and History-Dependent Consensus Endpoints. For a connected simple undirected graph with Laplacian $L$, maximum degree $d_{\max}$, and gain $0<c<d_{\max}^{-1}$, the matrix $P=I-cL$ is nonnegative, symmetric, doubly stochastic, and has positive diagonal. It preserves the nonnegative and strictly positive row-stochastic state spaces and their population mean. On the disagreement subspace it contracts geometrically in Frobenius norm with factor $\rho=\max_{j\ge 2}|1-c\lambda_j(L)|<1$. We give a self-contained proof, identify the exact spectral-contraction range $0<c<2/\lambda_n(L)$ when stochasticity is not required, show why the strict governing gain bound cannot be replaced uniformly by $c\le d_{\max}^{-1}$, exhibit the attainable case $\rho=0$, and separate positive-state invariance from positive-diagonal and contraction requirements. These statements concern one fixed connected averaging epoch only; they do not establish results for delayed, asynchronous, open-membership, or distributed membership-transition dynamics.
Keywords: average consensus · graph Laplacian · spectral contraction · stochastic matrix · simplex invariance
Guillaume Bolduc · Preprint · 2026-09-03 · v07
Abstract
We characterize the mean priority returned by a single-agent arrival--mixing--departure history. Each row is a probability vector over the active objectives, so removing an agent also removes a coordinate and conditions the surviving rows. For arbitrary nonnegative row-stochastic mixing, we derive the returned mean directly from the incumbent state, retention masses and mixing matrix. Holding those inputs fixed gives the unique restoring newcomer conditional priority when its influence is positive, an exact strict-positivity feasibility test, and a separate zero-influence case. Under equal retention and doubly stochastic mixing, restoration requires a weighted average of incumbent rows; for one unweighted consensus update, this is the newcomer-neighbor row mean. The general result is an offline input-level characterization, not a distributed arrival protocol. An exact counterexample excludes unrestricted endpoint invariance. Standard final-epoch average consensus supplies the limiting-priority interpretation; no new physical-rendezvous or persistently open convergence theorem is claimed.
Keywords: priority consensus · open multi-agent systems · membership changes · stochastic matrices · simplex · multi-objective optimization
Shokoufeh Naderi, Maude J. Blondin, Sébastien Roy, Guillaume Bolduc* · Preprint · 2026-03-17 · arXiv v1
*Guillaume Bolduc: fourth-author credit pending update. The linked arXiv v1 currently lists Shokoufeh Naderi, Maude J. Blondin, and Sébastien Roy.
Abstract
This paper analyzes consensus in multi-agent systems under uniform and nonuniform communication delays, a key challenge in distributed coordination with applications to robotic swarms. It investigates the convergence of a consensus algorithm accounting for delays across communication links in a connected, undirected graph. Novel convergence results are derived using Rouché's theorem and Lyapunov-based stability analysis. The system is shown to reach consensus at a steady-state value given by a weighted average determined by the delay distribution, with stability ensured under explicit parameter bounds. Both uniform and nonuniform delay scenarios are analyzed, and the corresponding convergence values are explicitly derived. The theoretical results are validated through simulations, which explore the impact of delay heterogeneity on consensus outcomes. Furthermore, the algorithm is implemented and experimentally tested on a swarm of QBOT3 ground robots to solve the rendezvous problem, demonstrating the agents’ ability to converge to a common location despite realistic communication constraints, thus confirming the algorithm’s robustness and practical applicability. The results provide guidelines for designing consensus protocols that tolerate communication delays, offer insights into the relationship between network delays and coordination performance, and demonstrate their applicability to distributed robotic systems.
Keywords: multi-agent consensus · uniform communication delays · nonuniform communication delays · multi-objective optimization · robotic swarms